Introduction to Regularity · PDF fileINTRODUCTION 3 From a “philosophical”...

Introduction to Regularity Structures

March 27, 2014

Martin Hairer

The University of Warwick, Email: [email protected]

AbstractThese are short notes from a series of lectures given at the University of Rennes in June2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probabilityin Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a conciseoverview of the theory of regularity structures as exposed in the article [Hai14]. In order toallow to focus on the conceptual aspects of the theory, many proofs are omitted and statementsare simplified. We focus on applying the theory to the problem of giving a solution theory tothe stochastic quantisation equations for the Euclidean Φ4

3 quantum field theory.

Contents

1 Introduction 1

2 Definitions and the reconstruction operator 4

3 Examples of regularity structures 10

4 Products and composition by smooth functions 13

5 Schauder estimates and admissible models 16

6 Application of the theory to semilinear SPDEs 20

7 Renormalisation of the dynamical Φ43 model 23

1 Introduction

Very recently, a new theory of “regularity structures” was introduced [Hai14], unifying variousflavours of the theory of (controlled) rough paths (including Gubinelli’s theory of controlledrough paths [Gub04], as well as his branched rough paths [Gub10]), as well as the usual Taylorexpansions. While it has its roots in the theory of rough paths [Lyo98], the main advantageof this new theory is that it is no longer tied to the one-dimensionality of the time parameter,which makes it also suitable for the description of solutions to stochastic partial differentialequations, rather than just stochastic ordinary differential equations. The aim of this articleis to give a concise survey of the theory while focusing on the construction of the dynamicalΦ4

3 model. While the exposition aims to be reasonably self-contained (in particular no priorknowledge of the theory of rough paths is assumed), most of the proofs will only be sketched.

The main achievement of the theory of regularity structures is that it allows to give a(pathwise!) meaning to ill-posed stochastic PDEs that arise naturally when trying to describe

2 INTRODUCTION

the macroscopic behaviour of models from statistical mechanics near criticality. One exampleof such an equation is the KPZ equation arising as a natural model for one-dimensionalinterface motion [KPZ86, BG97, Hai13]:

∂th = ∂2xh+ (∂xh)2 + ξ − C .

Another example is the dynamical Φ43 model arising for example in the stochastic quantisation

of Euclidean quantum field theory [PW81, JLM85, AR91, DPD03, Hai14], as well as auniversal model for phase coexistence near the critical point [GLP99]:

∂tΦ = ∆Φ + CΦ− Φ3 + ξ .

In both of these examples, ξ formally denotes space-time white noise, C is an arbitrary constant(which will actually turn out to be infinite in some sense!), and we consider a bounded squarespatial domain with periodic boundary conditions. In the case of the dynamical Φ4

3 model, thespatial variable has dimension 3, while it has dimension 1 in the case of the KPZ equation.While a full exposition of the theory is well beyond the scope of this short introduction, weaim to give a concise overview to most of its concepts. In most cases, we will only stateresults in a rather informal way and give some ideas as to how the proofs work, focusing onconceptual rather than technical issues. The only exception is the “reconstruction theorem”,Theorem 2.10 below, which is the linchpin of the whole theory. Since its proof (or rather aslightly simplified version of it) is relatively concise, we provide a fully self-contained version.For precise statements and complete proofs of most of the results exposed here, we refer to theoriginal article [Hai14].

Loosely speaking, the type of well-posedness results that can be proven with the help ofthe theory of regularity structures can be formulated as follows.

Theorem 1.1 Let ξε = δε ∗ ξ denote the regularisation of space-time white noise with acompactly supported smooth mollifier δε that is scaled by ε in the spatial direction(s) and byε2 in the time direction. Denote by hε and Φε the solutions to

∂thε = ∂2xhε + (∂xhε)2 − Cε + ξε ,

∂tΦε = ∆Φε + CεΦε − Φ3ε + ξε .

Then, there exist choices of constants Cε and Cε diverging as ε→ 0, as well as processes hand Φ such that hε → h and Φε → Φ in probability. Furthermore, while the constants Cε andCε do depend crucially on the choice of mollifiers δε, the limits h and Φ do not depend onthem.

Remark 1.2 We made a severe abuse of notation here since the space-time white noiseappearing in the equation for hε is on R× T1, while the one appearing in the equation for Φε

is on R× T3. (Here we denote by Tn the n-dimensional torus.)

Remark 1.3 We have not described the topology in which the convergence takes place inthese examples. In the case of the KPZ equation, one actually obtains convergence in somespace of space-time Holder continuous functions. In the case of the dynamical Φ4

3 model,convergence takes place in some space of space-time distributions. One caveat that also has tobe dealt with in the latter case is that the limiting process Φ may in principle explode in finitetime for some instances of the driving noise.

INTRODUCTION 3

From a “philosophical” perspective, the theory of regularity structures is inspired by thetheory of controlled rough paths [Lyo98, Gub04, LCL07], so let us rapidly survey the mainideas of that theory. The setting of the theory of controlled rough paths is the following. Let’ssay that we want to solve a controlled differential equation of the type

dY = f (Y ) dX(t) , (1.1)

where X ∈ Cα is a rather rough function (say a typical sample path for an m-dimensionalBrownian motion). It is a classical result by Young [You36] that the Riemann-Stieltjes integral(X,Y ) 7→

∫ ·0 Y dX makes sense as a continuous map from Cα × Cα into Cα if and only if

α > 12 . As a consequence, “naıve” approaches to a pathwise solution to (1.1) are bound to fail

if X has the regularity of Brownian motion.The main idea is to exploit the a priori “guess” that solutions to (1.1) should “look like X

at small scales”. More precisely, one would naturally expect the solution Y to satisfy

Yt = Ys + Y ′sXs,t +O(|t− s|2α) , (1.2)

where we wrote Xs,t as a shorthand for the increment Xt−Xs. As a matter of fact, one wouldexpect to have such an expansion with Y ′ = f (Y ). Denote by CαX the space of pairs (Y, Y ′)satisfying (1.2) for a given “model path” X . It is then possible to simply “postulate” the valuesof the integrals

Xs,t =:

∫ t

sXs,r ⊗ dXr , (1.3)

satisfying “Chen’s relations”

Xs,t − Xs,u − Xu,t = Xs,u ⊗Xu,t , (1.4)

as well as the analytic bound |Xs,t| . |t − s|2α, and to exploit this additional data to give acoherent definition of expressions of the type

∫Y dX , provided that the path X is “enhanced”

with its iterated integrals X and Y is a “controlled path” of the type (1.2). See for example[Gub04] for more information or [Hai11] for a concise exposition of this theory.

Compare (1.2) to the fact that a function f : R→ R is of class Cγ with γ ∈ (k, k + 1) iffor every s ∈ R there exist coefficients f (1)

s , . . . , f (k)s such that

ft = fs +

k∑`=1

f (`)s (t− s)` +O(|t− s|γ) . (1.5)

Of course, f (`)s is nothing but the `th derivative of f at the point s, divided by `!. In this sense,

one should really think of a controlled rough path (Y, Y ′) ∈ CαX as a 2α-Holder continuousfunction, but with respect to a “model” determined by the function X , rather than by theusual Taylor polynomials. This formal analogy between controlled rough paths and Taylorexpansions suggests that it might be fruitful to systematically investigate what are the “right”objects that could possibly take the place of Taylor polynomials, while still retaining many oftheir nice properties.

AcknowledgementsFinancial support from the Leverhulme trust through a leadership award is gratefully acknowledged.

4 DEFINITIONS AND THE RECONSTRUCTION OPERATOR

2 Definitions and the reconstruction operator

The first step in such an endeavour is to set up an algebraic structure reflecting the propertiesof Taylor expansions. First of all, such a structure should contain a vector space T that willcontain the coefficients of our expansion. It is natural to assume that T has a graded structure:T =

⊕α∈A Tα, for some set A of possible “homogeneities”. For example, in the case of the

usual Taylor expansion (1.5), it is natural to take for A the set of natural numbers and to haveT` contain the coefficients corresponding to the derivatives of order `. In the case of controlledrough paths however, it is natural to take A = {0, α}, to have again T0 contain the value ofthe function Y at any time s, and to have Tα contain the Gubinelli derivative Y ′s . This reflectsthe fact that the “monomial” t 7→ Xs,t only vanishes at order α near t = s, while the usualmonomials t 7→ (t− s)` vanish at integer order `.

This however isn’t the full algebraic structure describing Taylor-like expansions. Indeed,one of the characteristics of Taylor expansions is that an expansion around some point x0 canbe re-expanded around any other point x1 by writing

(x− x0)m =∑

k+`=m

m!

k!`!(x1 − x0)k · (x− x1)` . (2.1)

(In the case when x ∈ Rd, k, ` and m denote multi-indices and k! = k1! . . . kd!.) Somewhatsimilarly, in the case of controlled rough paths, we have the (rather trivial) identity

Xs0,t = Xs0,s1 · 1 + 1 ·Xs1,t . (2.2)

What is a natural abstraction of this fact? In terms of the coefficients of a “Taylor expansion”,the operation of reexpanding around a different point is ultimately just a linear operationfrom Γ: T → T , where the precise value of the map Γ depends on the starting point x0, theendpoint x1, and possibly also on the details of the particular “model” that we are considering.In view of the above examples, it is natural to impose furthermore that Γ has the propertythat if τ ∈ Tα, then Γτ − τ ∈

⊕β<α Tβ . In other words, when reexpanding a homogeneous

monomial around a different point, the leading order coefficient remains the same, but lowerorder monomials may appear.

These heuristic considerations can be summarised in the following definition of an abstractobject we call a regularity structure:

Definition 2.1 Let A ⊂ R be bounded from below and without accumulation point, and letT =

⊕α∈A Tα be a vector space graded by A such that each Tα is a Banach space. Let

furthermore G be a group of continuous operators on T such that, for every α ∈ A, everyΓ ∈ G, and every τ ∈ Tα, one has Γτ − τ ∈

⊕β<α Tβ . The triple T = (A, T,G) is called a

regularity structure with model space T and structure group G.

Remark 2.2 Given τ ∈ T , we will write ‖τ‖α for the norm of its component in Tα.

Remark 2.3 In [Hai14] it is furthermore assumed that 0 ∈ A, T0 ≈ R, and T0 is invariantunder G. This is a very natural assumption which ensures that our regularity structure is atleast sufficiently rich to represent constant functions.

DEFINITIONS AND THE RECONSTRUCTION OPERATOR 5

Remark 2.4 In principle, the set A can be infinite. By analogy with the polynomials, it is thennatural to consider T as the set of all formal series of the form

∑α∈A τα, where only finitely

many of the τα’s are non-zero. This also dovetails nicely with the particular form of elementsin G. In practice however we will only ever work with finite subsets of A so that the precisetopology on T does not matter.

At this stage, a regularity structure is a completely abstract object. It only becomes usefulwhen endowed with a model, which is a concrete way of associating to any τ ∈ T and x0 ∈ Rd,the actual “Taylor polynomial based at x0” represented by τ . Furthermore, we want elementsτ ∈ Tα to represent functions (or possibly distributions!) that “vanish at order α” around thegiven point x0.

Since we would like to allow A to contain negative values and therefore allow elements inT to represent actual distributions, we need a suitable notion of “vanishing at order α”. Weachieve this by considering the size of our distributions, when tested against test functionsthat are localised around the given point x0. Given a test function ϕ on Rd, we write ϕλx as ashorthand for

ϕλx(y) = λ−dϕ(λ−1(y − x)) .

Given r > 0, we also denote by Br the set of all functions ϕ : Rd → R such that ϕ ∈ Cr with‖ϕ‖Cr ≤ 1 that are furthermore supported in the unit ball around the origin. With this notation,our definition of a model for a given regularity structure T is as follows.

Definition 2.5 Given a regularity structure T and an integer d ≥ 1, a model for T on Rd

consists of maps

Π: Rd → L(T,S ′(Rd)) Γ: Rd × Rd → G

x 7→ Πx (x, y) 7→ Γxy

such that ΓxyΓyz = Γxz and ΠxΓxy = Πy. Furthermore, given r > | infA|, for any compactset K ⊂ Rd and constant γ > 0, there exists a constant C such that the bounds

|(Πxτ)(ϕλx)| ≤ Cλ|τ |‖τ‖α , ‖Γxyτ‖β ≤ C|x− y|α−β‖τ‖α , (2.3)

hold uniformly over ϕ ∈ Br, (x, y) ∈ K, λ ∈ (0, 1], τ ∈ Tα with α ≤ γ, and β < α.

Remark 2.6 In principle, test functions appearing in (2.3) should be smooth. It turns out thatif these bounds hold for smooth elements of Br, then Πxτ can be extended canonically to allowany Cr test function with compact support.

Remark 2.7 The identity ΠxΓxy = Πy reflects the fact that Γxy is the linear map that takesan expansion around y and turns it into an expansion around x. The first bound in (2.3) stateswhat we mean precisely when we say that τ ∈ Tα represents a term that vanishes at order α.(Note that α can be negative, so that this may actually not vanish at all!) The second bound in(2.3) is very natural in view of both (2.1) and (2.2). It states that when expanding a monomialof order α around a new point at distance h from the old one, the coefficient appearing in frontof lower-order monomials of order β is of order at most hα−β .


Remark 2.8 In many cases of interest, it is natural to scale the different directions of Rd in adifferent way. This is the case for example when using the theory of regularity structures tobuild solution theories for parabolic stochastic PDEs, in which case the time direction “countsas” two space directions. To deal with such a situation, one can introduce a scaling s of Rd,which is just a collection of d mutually prime strictly positive integers and to define ϕλx in sucha way that the ith direction is scaled by λsi . In this case, the Euclidean distance between twopoints should be replaced everywhere by the corresponding scaled distance |x|s =

∑i |xi|1/si .

See also [Hai14] for more details.

With these definitions at hand, it is then natural to define an equivalent in this context ofthe space of γ-Holder continuous functions in the following way.

Definition 2.9 Given a regularity structure T equipped with a model (Π,Γ) over Rd, thespaceDγ = Dγ(T ,Γ) is given by the set of functions f : Rd →

⊕α<γ Tα such that, for every

compact set K and every α < γ, the exists a constant C with

‖f (x)− Γxyf (y)‖α ≤ C|x− y|γ−α (2.4)

uniformly over x, y ∈ K.

The most fundamental result in the theory of regularity structures then states that givenf ∈ Dγ with γ > 0, there exists a unique Schwartz distributionRf on Rd such that, for everyx ∈ Rd,Rf “looks like Πxf (x) near x”. More precisely, one has

Theorem 2.10 Let T be a regularity structure as above and let (Π,Γ) a model for T on Rd.Then, there exists a unique linear mapR : Dγ → S ′(Rd) such that

|(Rf −Πxf (x))(ϕλx)| . λγ , (2.5)

uniformly over ϕ ∈ Br and λ as before, and locally uniformly in x.

Proof. The proof of the theorem relies on the following fact. Given any r > 0 (but finite!),there exists a function ϕ : Rd → R with the following properties:

(1) The function ϕ is of class Cr and has compact support.(2) For every polynomial P of degree r, there exists a polynomial P of degree r such that,

for every x ∈ Rd, one has∑

y∈Zd P (y)ϕ(x− y) = P (x).

(3) One has∫ϕ(x)ϕ(x− y) dx = δy,0 for every y ∈ Zd.

(4) There exist coefficients {ak}k∈Zd such that 2−d/2ϕ(x/2) =∑

k∈Zd akϕ(x− k).The existence of such a function ϕ is highly non-trivial. This is actually equivalent to theexistence of a wavelet basis consisting of Cr functions with compact support, a proof of whichwas first obtained by Daubechies in her seminal article [Dau88]. From now on, we take theexistence of such a function ϕ as a given for some r > | infA|. We also set Λn = 2−nZd and,for y ∈ Λn, we set ϕny (x) = 2nd/2ϕ(2n(x−y)). Here, the normalisation is chosen in such a waythat the set {ϕny}y∈Λn is again orthonormal in L2. We then denote by Vn ⊂ Cr the linear spanof {ϕny}y∈Λn , so that, by the property (4) above, one has V0 ⊂ V1 ⊂ V2 ⊂ . . .. We furthermoredenote by Vn the L2-orthogonal complement of Vn−1 in Vn, so that Vn = V0 ⊕ V1 ⊕ . . .⊕ Vn.In order to keep notations compact, it will also be convenient to define the coefficients ank withk ∈ Λn by ank = a2nk.


With these notations at hand, we then define a sequence of linear operatorsRn : Dγ → Crby

(Rnf)(y) =∑x∈Λn

(Πxf (x))(ϕnx)ϕnx(y) .

We claim that there then exists a Schwartz distribution Rf such that, for every compactlysupported test function ψ of class Cr, one has 〈Rnf, ψ〉 → (Rf)(ψ), and thatRf furthermoresatisfies the properties stated in the theorem.

Let us first consider the size of the components of Rn+1f −Rnf in Vn. Given x ∈ Λn,we make use of properties (3-4), so that

〈Rn+1f −Rnf, ϕnx〉 =∑

k∈Λn+1

ank〈Rn+1f, ϕn+1x+k〉 − (Πxf (x))(ϕnx)

=∑

k∈Λn+1

ank(Πx+kf (x+ k))(ϕn+1x+k)− (Πxf (x))(ϕnx)

=∑

k∈Λn+1

ank((Πx+kf (x+ k))(ϕn+1x+k)− (Πxf (x))(ϕn+1

x+k))

=∑

k∈Λn+1

ank(Πx+k(f (x+ k)− Γx+k,xf (x)))(ϕn+1x+k) ,

where we used the algebraic relations between Πx and Γxy to obtain the last identity. Sinceonly finitely many of the coefficients ak are non-zero, it follows from the definition of Dγ thatfor the non-vanishing terms in this sum we have the bound

‖f (x+ k)− Γx+k,xf (x)‖α . 2−n(γ−α) ,

uniformly over n ≥ 0 and x in any compact set. Furthermore, for any τ ∈ Tα, it follows fromthe definition of a model that one has the bound

|(Πxτ)(ϕnx)| . 2−αn−nd2 ,

again uniformly over n ≥ 0 and x in any compact set. Here, the additional factor 2−nd2 comes

from the fact that the functions ϕnx are normalised in L2 rather than L1. Combining these twobounds, we immediately obtain that

|〈Rn+1f −Rnf, ϕnx〉| . 2−γn−nd2 , (2.6)

uniformly over n ≥ 0 and x in compact sets. Take now a test function ψ ∈ Cr with compactsupport and let us try to estimate 〈Rn+1f − Rnf, ψ〉. Since Rn+1f − Rnf ∈ Vn+1, wecan decompose it into a part δRnf ∈ Vn and a part δRnf ∈ Vn+1 and estimate both partsseparately. Regarding the part in Vn, we have

|〈δRnf, ψ〉| =∣∣∣ ∑x∈Λn+1

〈δRnf, ϕnx〉〈ϕnx, ψ〉∣∣∣ . 2−γn−

nd2

∑x∈Λn+1

|〈ϕnx, ψ〉| , (2.7)

where we made use of the bound (2.6). At this stage we use the fact that, due to the boundednessof ψ, we have |〈ϕnx, ψ〉| . 2−nd/2. Furthermore, thanks to the boundedness of the support ofψ, the number of non-vanishing terms appearing in this sum is bounded by 2nd, so that weeventually obtain the bound

|〈δRnf, ψ〉| . 2−γn . (2.8)


Regarding the second term, we use the standard fact coming from wavelet analysis [Mey92]that a basis of Vn+1 can be obtained in the same way as the basis of Vn, but replacing thefunction ϕ by functions ϕ from some finite set Φ. In other words, Vn+1 is the linear span of{ϕnx}x∈Λn;ϕ∈Φ. Furthermore, as a consequence of property (2), the functions ϕ ∈ Φ all havethe property that ∫

ϕ(x)P (x) dx = 0 , (2.9)

for any polynomial P of degree less or equal to r. In particular, this shows that one has thebound

|〈ϕnx, ψ〉| . 2−nd2−nr .

As a consequence, one has

|〈δRnf, ψ〉| =∣∣∣ ∑x∈Λn

ϕ∈Φ

〈Rn+1f, ϕnx〉〈ϕnx, ψ〉∣∣∣ . 2−

nd2−nr∣∣∣ ∑x∈Λn

ϕ∈Φ

〈Rn+1f, ϕnx〉∣∣∣ .

At this stage, we note that, thanks to the definition ofRn+1 and the bounds on the model (Π,Γ),we have |〈Rn+1f, ϕnx〉| . 2−

nd2−α0n, where α0 = infA, so that |〈δRnf, ψ〉| . 2−nr−α0n.

Combining this with (2.8), we see that one has indeed Rnf → Rf for some SchwartzdistributionRf .

It remains to show that the bound (2.5) holds. For this, given a distribution η ∈ Cα forsome α > −r, we first introduce the notation

Pnη =∑x∈Λn

η(ϕnx)ϕnx , Pnη =∑ϕ∈Φ

∑x∈Λn

η(ϕnx) ϕnx .

We also choose an integer value n ≥ 0 such that 2−n ∼ λ and we write

Rf −Πxf (x) = Rnf − PnΠxf (x) +∑m≥n

(Rm+1f −Rmf − PmΠxf (x))

= Rnf − PnΠxf (x) +∑m≥n

(δRmf − PmΠxf (x)) +∑m≥n

δRmf . (2.10)

We then test these terms against ψλx and we estimate the resulting terms separately. For thefirst term, we have the identity

(Rnf − PnΠxf (x))(ψλx) =∑y∈Λn

(Πyf (y)−Πxf (x))(ϕny ) 〈ϕny , ψλx〉 . (2.11)

We have the bound |〈ϕny , ψλx〉| . λ−d2−dn/2 ∼ 2dn/2. Since one furthermore has |y − x| . λfor all non-vanishing terms in the sum, one also has similarly to before

|(Πyf (y)−Πxf (x))(ϕny )| .∑α<γ

λγ−α2−dn2−αn ∼ 2−

dn2−γn . (2.12)

Since only finitely many (independently of n) terms contribute to the sum in (2.11), it is indeedbounded by a constant proportional to 2−γn ∼ λγ as required.


We now turn to the second term in (2.10), where we consider some fixed value m ≥ n. Werewrite this term very similarly to before as

(δRmf − PmΠxf (x))(ψλx) =∑ϕ∈Φ

∑y,z

(Πyf (y)−Πxf (x))(ϕm+1y ) 〈ϕm+1

y , ϕmz 〉〈ϕmz , ψλx〉 ,

where the sum runs over y ∈ Λm+1 and z ∈ Λm. This time, we use the fact that by the property(2.9) of the wavelets ϕ, one has the bound

|〈ϕmz , ψλx〉| . λ−d−r2−rm−md2 , (2.13)

and the L2-scaling implies that |〈ϕm+1y , ϕmz 〉| . 1. Furthermore, for each z ∈ Λm, only finitely

many elements y ∈ Λm+1 contribute to the sum, and these elements all satisfy |y − z| . 2−m.Bounding the first factor as in (2.12) and using the fact that there are of the order of λd2md

terms contributing for every fixed m, we thus see that the contribution of the second term in(2.10) is bounded by∑

m≥nλd2md

∑α<γ

λγ−α−d−r2−dm−αm−rm ∼∑α<γ

λγ−α−r∑m≥n

2−αm−rm ∼ λγ .

For the last term in (2.10), we combine (2.7) with the bound |〈ϕmy , ψλx〉| . λ−d2−dm/2 andthe fact that there are of the order of λd2−md terms appearing in the sum (2.7) to conclude thatthe mth summand is bounded by a constant proportional to 2−γm. Summing over m yieldsagain the desired bound and concludes the proof.

Remark 2.11 Note that the space Dγ depends crucially on the choice of model (Π,Γ). As aconsequence, the reconstruction operatorR itself also depends on that choice. However, themap (Π,Γ, f ) 7→ Rf turns out to be locally Lipschitz continuous provided that the distancebetween (Π,Γ, f ) and (Π, Γ, f ) is given by the smallest constant % such that

‖f (x)− f (x)− Γxyf (y) + Γxyf (y)‖α ≤ %|x− y|γ−α ,

|(Πxτ − Πxτ)(ϕλx)| ≤ %λα‖τ‖ ,

‖Γxyτ − Γxyτ‖β ≤ %|x− y|α−β‖τ‖ .

Here, in order to obtain bounds on (Rf − Rf)(ψ) for some smooth compactly supported testfunction ψ, the above bounds should hold uniformly for x and y in a neighbourhood of thesupport of ψ. The proof that this stronger continuity property also holds is actually crucialwhen showing that sequences of solutions to mollified equations all converge to the samelimiting object. However, its proof is somewhat more involved which is why we chose not togive it here.

Remark 2.12 In the particular case where Πxτ happens to be a continuous function for everyτ ∈ T (and every x ∈ Rd),Rf is also a continuous function and one has the identity

(Rf)(x) = (Πxf (x))(x) . (2.14)

This can be seen from the fact that

(Rf)(y) = limn→∞

(Rnf)(y) = limn→∞

∑x∈Λn

(Πxf (x))(ϕnx)ϕnx(y) .

10 EXAMPLES OF REGULARITY STRUCTURES

Indeed, our assumptions imply that the function (x, z) 7→ (Πxf (x))(z) is jointly continu-ous and since the non-vanishing terms in the above sum satisfy |x − y| . 2−n, one has2dn/2(Πxf (x))(ϕnx) ≈ (Πyf (y))(y) for large n. Since furthermore

∑x∈Λn ϕnx(y) = 2dn/2, the

claim follows.

3 Examples of regularity structures

3.1 The polynomial structure

It should by now be clear how the structure given by the usual Taylor polynomials fits into thisframework. A natural way of setting it up is to take for T the space of all abstract polynomialsin d commuting variables, denoted by X1, . . . , Xd, and to postulate that Tk consists of thelinear span of monomials of degree k. As an abstract group, the structure group G is then givenby Rd endowed with addition as its group operation, which acts onto T via ΓhX

k = (X −h)k,where h ∈ Rd and we use the notation Xk as a shorthand for Xk1

1 · · ·Xkdd for any multiindex

k.The canonical polynomial model is then given by

(ΠxXk)(y) = (y − x)k , Γxy = Γy−x .

We leave it as an exercise to the reader to verify that this does indeed satisfy the bounds andrelations of Definition 2.5.

In the particular case of the canonical polynomial model and for γ 6∈ N, the spacesDγ thencoincide precisely with the usual Holder spaces Cγ . In the case of integer values, this shouldbe interpreted as bounded functions for γ = 0, Lipschitz continuous functions for γ = 1, etc.

3.2 Controlled rough paths

Let us see now how the theory of controlled rough paths can be reinterpreted in the light ofthis theory. For given α ∈ (1

3 ,12 ) and n ≥ 1, we can define a regularity structure T by setting

A = {α − 1, 2α − 1, 0, α}. We furthermore take for T0 a copy of R with unit vector 1, forTα and Tα−1 a copy of Rn with respective unit vectors Wj and Ξj , and for T2α−1 a copy ofRn×n with unit vectors WjΞi. The structure group G is taken to be isomorphic to Rn and, forx ∈ Rn, it acts on T via

Γx1 = 1 , ΓxΞi = Ξi , ΓxWi = Wi − xi1 , Γx(WjΞi) = WjΞi − xjΞi .

Let now X = (X,X) be an α-Holder continuous rough path with values in Rn. In other words,the functions X and X are as in the introduction, satisfying the relation (1.4) and the analyticbounds |Xt −Xs| . |t− s|α, |Xs,t| . |t− s|2α. It turns out that this defines a model for Tin the following way (recall that Xs,t is a shorthand for Xt −Xs):

Lemma 3.1 Given an α-Holder continuous rough path X, one can define a model for T onR by setting Γsu = ΓXs,u and

(Πs1)(t) = 1 , (ΠsWj)(t) = Xjs,t

(ΠsΞj)(ψ) =

∫ψ(t) dXj

t , (ΠsWjΞi)(ψ) =

∫ψ(t) dXi,js,t .

EXAMPLES OF REGULARITY STRUCTURES 11

Here, both integrals are perfectly well-defined Riemann integrals, with the differential in thesecond case taken with respect to the variable t. Given a controlled rough path (Y, Y ′) ∈ CαXas in (1.2), this then defines an element Y ∈ D2α by setting

Y (s) = Y (s) 1 + Y ′i (s)Wi ,

with summation over i implied.

Proof. We first check that the algebraic properties of Definition 2.5 are satisfied. It is clearthat ΓsuΓut = Γst and that ΠsΓsuτ = Πuτ for τ ∈ {1,Wj ,Ξj}. Regarding WjΞi, wedifferentiate Chen’s relations (1.4) which yields the identity

dXi,js,t = dXi,ju,t +Xis,u dX

jt .

The last missing algebraic relation then follows at once. The required analytic bounds followimmediately from the definition of the rough path space Dα.

Regarding the function Y defined in the statement, we have

‖Y (s)− ΓsuY (u)‖0 = |Y (s)− Y (u) + Y ′i (u)Xis,u| ,

‖Y (s)− ΓsuY (u)‖α = |Y ′(s)− Y ′(u)| ,

so that the condition (2.4) with γ = 2α does indeed coincide with the definition of a controlledrough path given in the introduction.

In this context, the reconstruction theorem allows us to define an integration operator withrespect toW . We can formulate this as follows where one should really think of Z as providinga consistent definition of what one means by

∫Y dXj .

Lemma 3.2 In the same context as above, let α ∈ (13 ,

12 ), and consider Y ∈ D2α built as

above from a controlled rough path. Then, the map Y Ξi given by

(Y Ξj)(s) = Y (s) Ξj + Y ′i (s)WiΞj

belongs to D3α−1. Furthermore, there exists a function Z such that, for every smooth testfunction ψ, one has

(RY Ξj)(ψ) =

∫ψ(t) dZ(t) ,

and such that Zs,t = Y (s)Xjs,t + Y ′i (s)Xi,js,t +O(|t− s|3α).

Proof. The fact that Y Ξi ∈ D3α−1 is an immediate consequence of the definitions. Sinceα > 1

3 by assumption, we can apply the reconstruction theorem to it, from which it followsthat there exists a unique distribution η such that, if ψ is a smooth compactly supported testfunction, one has

η(ψλs ) =

∫ψλs (t)Y (s) dXj

t +

∫ψλs (t)Y ′i (s) dXi,js,t +O(λ3α−1) .

By a simple approximation argument, it turns out that one can take for ψ the indicator functionof the interval [0, 1], so that

η(1[s,t]) = Y (s)Xjs,t + Y ′i (s)Xi,js,t +O(|t− s|3α) .

Here, the reason why one obtains an exponent 3α rather than 3α − 1 is that it is really|t− s|−11[s,t] that scales like an approximate δ-distribution as t→ s.

12 EXAMPLES OF REGULARITY STRUCTURES

Remark 3.3 Using the formula (2.14), it is straightforward to verify that if X happens to be asmooth function and X is defined from X via (1.3), but this time viewing it as a definition forthe left hand side, with the right hand side given by a usual Riemann integral, then the functionZ constructed in Lemma 3.2 coincides with the usual Riemann integral of Y against Xj .

3.3 A classical result from harmonic analysisThe considerations above suggest that a very natural space of distributions is obtained in thefollowing way. For some α > 0, we denote by C−α the space of all Schwartz distributions ηsuch that η belongs to the dual of Cr with r > α some integer and such that

|η(ϕλx)| . λ−α ,

uniformly over all ϕ ∈ Br and λ ∈ (0, 1], and locally uniformly in x. Given any compact setK, the best possible constant such that the above bound holds uniformly over x ∈ K yields aseminorm. The collection of these seminorms endows C−α with a Frechet space structure.

Remark 3.4 It turns out that the space C−α is independent of the choice of r in the definitiongiven above, which justifies the notation. Different values of r give raise to equivalentseminorms.

Remark 3.5 In terms of the scale of classical Besov spaces, the space C−α is a local versionof B−α∞,∞. It is in some sense the largest space of distributions that is invariant under the scalingϕ(·) 7→ λ−αϕ(λ−1·), see for example [BP08].

It is then a classical result in the “folklore” of harmonic analysis that the product extendsnaturally to C−α × Cβ into S ′(Rd) if and only if β > α. The reconstruction theorem yields astraightforward proof of the “if” part of this result:

Theorem 3.6 There is a continuous bilinear mapB : C−α×Cβ → S ′(Rd) such thatB(f, g) =fg for any two continuous functions f and g.

Proof. Assume from now on that ξ ∈ C−α for some α > 0 and that f ∈ Cβ for someβ > α. We then build a regularity structure T in the following way. For the set A, we takeA = N ∪ (N− α) and for T , we set T = V ⊕W , where each one of the spaces V and W is acopy of the polynomial model in d commuting variables constructed in Section 3.1. We alsochoose Γ as in the canonical model, acting simultaneously on each of the two instances.

As before, we denote by Xk the canonical basis vectors in V . We also use the suggestivenotation “ΞXk” for the corresponding basis vector in W , but we postulate that ΞXk ∈ Tα+|k|rather than ΞXk ∈ T|k|. Given any distribution ξ ∈ C−α, we then define a model (Πξ,Γ),where Γ is as in the canonical model, while Πξ acts as

(ΠξxX

k)(y) = (y − x)k , (ΠξxΞXk)(y) = (y − x)kξ(y) ,

with the obvious abuse of notation in the second expression. It is then straightforward to verifythat Πy = Πx ◦ Γxy and that the relevant analytical bounds are satisfied, so that this is indeeda model.

Denote now byRξ the reconstruction map associated to the model (Πξ,Γ) and, for f ∈ Cβ ,denote by F the element in Dβ given by the local Taylor expansion of f of order β at each

PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS 13

point. Note that even though the space Dβ does in principle depend on the choice of model,in our situation F ∈ Dβ for any choice of ξ. It follows immediately from the definitionsthat the map x 7→ ΞF (x) belongs to Dβ−α so that, provided that β > α, one can apply thereconstruction operator to it. This suggests that the multiplication operator we are looking forcan be defined as

B(f, ξ) = Rξ(ΞF ) .

By Theorem 2.10, this is a jointly continuous map from Cβ × C−α into S ′(Rd), provided thatβ > α. If ξ happens to be a smooth function, then it follows immediately from Remark 2.12that B(f, ξ) = f (x)ξ(x), so that B is indeed the requested continuous extension of the usualproduct.

Remark 3.7 As a consequence of (2.5), it is actually easy to show that B : C−α × Cβ → C−α.

4 Products and composition by smooth functions

One of the main purposes of the theory presented here is to give a robust way to multiplydistributions (or functions with distributions) that goes beyond the barrier illustrated by Theo-rem 3.6. Provided that our functions / distributions are represented as elements in Dγ for somemodel and regularity structure, we can multiply their “Taylor expansions” pointwise, providedthat we give ourselves a table of multiplication on T .

It is natural to consider products with the following properties. Here, given a regularitystructure, we say that a subspace V ⊂ T is a sector if it is invariant under the action of thestructure group G and if it can furthermore be written as V =

⊕α∈A Vα with Vα ⊂ Tα.

Definition 4.1 Given a regularity structure (T,A,G) and two sectors V, V ⊂ T , a producton (V, V ) is a bilinear map ? : V × V → T such that, for any τ ∈ Vα and τ ∈ Vβ , one hasτ ? τ ∈ Tα+β and such that, for any element Γ ∈ G, one has Γ(τ ? τ ) = Γτ ? Γτ .

Remark 4.2 The condition that homogeneities add up under multiplication is very naturalbearing in mind the case of the polynomial regularity structure. The second condition is alsovery natural since it merely states that if one reexpands the product of two “polynomials”around a different point, one should obtain the same result as if one reexpands each factor firstand then multiplies them together.

Given such a product, we can ask ourselves when the pointwise product of an elementDγ1 with an element in Dγ2 again belongs to some Dγ . In order to answer this question, weintroduce the notation Dγα to denote those elements f ∈ Dγ such that furthermore

f (x) ∈ T+α ≡

⊕β≥α

Tβ ,

for every x. With this notation at hand, it is not too difficult to verify that one has the followingresult:

Theorem 4.3 Let f1 ∈ Dγ1α1(V ), f2 ∈ Dγ2

α2(V ), and let ? be a product on (V, V ). Then, thefunction f given by f (x) = f1(x) ? f2(x) belongs to Dγα with

α = α1 + α2 , γ = (γ1 + α2) ∧ (γ2 + α1) . (4.1)

14 PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS

Proof. It is clear that f (x) ∈⊕

β>α Tβ , so it remains to show that it belongs to Dγ . Fur-thermore, since we are only interested in showing that f1 ? f2 ∈ Dγ , we discard all of thecomponents in Tβ for β ≥ γ.

By the properties of the product ?, it remains to obtain a bound of the type

‖Γxyf1(y) ? Γxyf2(y)− f1(x) ? f2(x)‖β . |x− y|γ−β .

By adding and subtracting suitable terms, we obtain

‖Γxyf (y)− f (x)‖β ≤ ‖(Γxyf1(y)− f1(x)) ? (Γxyf2(y)− f2(x))‖β (4.2)

+ ‖(Γxyf1(y)− f1(x)) ? f2(x)‖β + ‖f1(x) ? (Γxyf2(y)− f2(x))‖β .

It follows from the properties of the product ? that the first term in (4.2) is bounded by aconstant times∑

β1+β2=β

‖Γxyf1(y)− f1(x)‖β1‖Γxyf2(y)− f2(x)‖β2

.∑

β1+β2=β

‖x− y‖γ1−β1‖x− y‖γ2−β2 . ‖x− y‖γ1+γ2−β .

Since γ1 + γ2 ≥ γ, this bound is as required. The second term is bounded by a constant times∑β1+β2=β

‖Γxyf1(y)− f1(x)‖β1‖f2(x)‖β2 . ‖x− y‖γ1−β1 1β2≥α2 . ‖x− y‖γ1+α2−β ,

where the second inequality uses the identity β1 + β2 = β. Since γ1 + α2 ≥ γ, this bound isagain of the required type. The last term is bounded similarly by reversing the roles played byf1 and f2.

Remark 4.4 It is clear that the formula (4.1) for γ is optimal in general as can be seen fromthe following two “reality checks”. First, consider the case of the polynomial model and takefi ∈ Cγi . In this case, the truncated Taylor series Fi for fi belong to Dγi0 . It is clear that in thiscase, the product cannot be expected to have better regularity than γ1 ∧ γ2 in general, whichis indeed what (4.1) states. The second reality check comes from the example of Section 3.3.In this case, one has F ∈ Dβ0 , while the constant function x 7→ Ξ belongs to D∞−α so that,according to (4.1), one expects their product to belong to Dβ−α−α , which is indeed the case.

It turns out that if we have a product on a regularity structure, then in many cases this alsonaturally yields a notion of composition with smooth functions. Of course, one could in generalnot expect to be able to compose a smooth function with a distribution of negative order. As amatter of fact, we will only define the composition of smooth functions with elements in someDγ for which it is guaranteed that the reconstruction operator yields a continuous function.One might think at this case that this would yield a triviality, since we know of course howto compose arbitrary continuous function. The subtlety is that we would like to design ourcomposition operator in such a way that the result is again an element of Dγ .

For this purpose, we say that a given sector V ⊂ T is function-like if α < 0 ⇒ Vα = 0and if V0 is one-dimensional. (Denote the unit vector of V0 by 1.) We will furthermore alwaysassume that our models are normal in the sense that (Πx1)(y) = 1. I this case, it turns out that

PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS 15

if f ∈ Dγ(V ), thenRf is a continuous function and one has the identity (Rf)(x) = 〈1, f (x)〉,where we denote by 〈1, ·〉 the element in the dual of V which picks out the prefactor of 1.

Assume now that we are given a regularity structure with a function-like sector V and aproduct ? : V ×V → V . For any smooth function G : R→ R and any f ∈ Dγ(V ) with γ > 0,we can then define G(f ) to be the V -valued function given by

(G ◦ f)(x) =∑k≥0

G(k)(f (x))k!

f (x)?k ,

where we have setf (x) = 〈1, f (x)〉 , f (x) = f (x)− f (x)1 .

Here, G(k) denotes the kth derivative of G and τ?k denotes the k-fold product τ ? · · · ? τ . Wealso used the usual conventions G(0) = G and τ?0 = 1.

Note that as long as G is C∞, this expression is well-defined. Indeed, by assumption, thereexists some α0 > 0 such that f (x) ∈ T+

α0. By the properties of the product, this implies that

one has f (x)?k ∈ T+kα0

. As a consequence, when considering the component of G ◦ f in Tβfor β < γ, the only terms that give a contribution are those with k < γ/α0. Since we cannotpossibly hope in general that G ◦ f ∈ Dγ′ for some γ′ > γ, this is all we really need.

It turns out that if G is sufficiently regular, then the map f 7→ G ◦ f enjoys similarly nicecontinuity properties to what we are used to from classical Holder spaces. The following resultis the analogue in this context to the well-known fact that the composition of a Cγ functionwith a sufficiently smooth function G is again of class Cγ .

Proposition 4.5 In the same setting as above, provided that G is of class Ck with k > γ/α0,the map f 7→ G ◦ f is continuous from Dγ(V ) into itself. If k > γ/α0 + 1, then it is locallyLipschitz continuous.

The proof of this result can be found in [Hai14]. It is somewhat lengthy, but ultimatelyrather straightforward.

4.1 A simple exampleA very important remark is that even if bothRf1 andRf2 happens to be continuous functions,this does not in general imply that R(f1 ? f2)(x) = (Rf1)(x) (Rf2)(x)! For example, fixκ < 0 and consider the regularity structure given by A = (−2κ,−κ, 0), with each Tα being acopy of R given by T−nκ = 〈Ξn〉. We furthermore take for G the trivial group. This regularitystructure comes with an obvious product by setting Ξm ?Ξn = Ξm+n provided thatm+n ≤ 2.

Then, we could for example take as a model for T = (T,A,G):

(ΠxΞ0)(y) = 1 , (ΠxΞ)(y) = 0 , (ΠxΞ2)(y) = c , (4.3)

where c is an arbitrary constant. Let furthermore

F1(x) = f1(x)Ξ0 + f ′1(x)Ξ , F2(x) = f2(x)Ξ0 + f ′2(x)Ξ .

Since our group G is trivial, one has Fi ∈ Dγ provided that each of the fi belongs to Dγ andeach of the f ′i belongs to Dγ+κ. (And one has γ + κ < 1.) One furthermore has the identity(RFi)(x) = fi(x).

16 SCHAUDER ESTIMATES AND ADMISSIBLE MODELS

However, the pointwise product is given by

(F1 ? F2)(x) = f1(x)f2(x)Ξ0 + (f ′1(x)f2(x) + f ′2(x)f1(x))Ξ + f ′1(x)f ′2(x)Ξ2 ,

which by Theorem 4.3 belongs to Dγ−κ. Provided that γ > κ, one can then apply thereconstruction operator to this product and one obtains

R(F1 ? F2)(x) = f1(x)f2(x) + cf ′1(x)f ′2(x) ,

which is obviously different from the pointwise productRF1 · RF2.How should this be interpreted? For n > 0, we could have defined a model Π(n) by

(ΠxΞ0)(y) = 1 , (ΠxΞ)(y) =√

2c sin(nx) , (ΠxΞ2)(y) = 2c sin2(nx) .

Denoting byR(n) the corresponding reconstruction operator, we have the identity

(R(n)Fi)(x) = fi(x) +√

2cf ′i(x) sin(nx) ,

as well asR(n)(F1 ?F2) = R(n)F1 · R(n)F2. As a model, the model Π(n) actually converges tothe limiting model Π defined in (4.3). As a consequence of the continuity of the reconstructionoperator, this implies that

R(n)F1 · R(n)F2 = R(n)(F1 ? F2)→ R(F1 ? F2) 6= RF1 · RF2 ,

which is of course also easy to see “by hand”. This shows that in some cases, the “non-standard”models as in (4.3) can be interpreted as limits of “standard” models for which the usual rulesof calculus hold. Even this is however not always the case.

5 Schauder estimates and admissible models

One of the reasons why the theory of regularity structures is very successful at providingdetailed descriptions of the small-scale features of solutions to semilinear (S)PDEs is thatit comes with very sharp Schauder estimates. Recall that the classical Schauder estimatesstate that if K : Rd → R is a kernel that is smooth everywhere, except for a singularity at theorigin that is (approximately) homogeneous of degree β − d for some β > 0, then the operatorf 7→ K ∗ f maps Cα into Cα+β for every α ∈ R, except for those values for which α+ β ∈ N.(See for example [Sim97].)

It turns out that similar Schauder estimates hold in the context of general regularitystructures in the sense that it is in general possible to build an operator K : Dγ → Dγ+β

with the property that RKf = K ∗ Rf . Of course, such a statement can only be true if ourregularity structure contains not only the objects necessary to describeRf up to order γ, butalso those required to describe K ∗ Rf up to order γ + β. What are these objects? At thisstage, it might be useful to reflect on the effect of the convolution of a singular function (ordistribution) with K.

Let us assume for a moment that f is also smooth everywhere, except at some point x0. Itis then straightforward to convince ourselves that K ∗ f is also smooth everywhere, except atx0. Indeed, for any δ > 0, we can write K = Kδ + Kc

δ , where Kδ is supported in a ball ofradius δ around 0 and Kc

δ is a smooth function. Similarly, we can decompose f as f = fδ +f cδ ,where fδ is supported in a δ-ball around x0 and f cδ is smooth. Since the convolution of a

SCHAUDER ESTIMATES AND ADMISSIBLE MODELS 17

smooth function with an arbitrary distribution is smooth, it follows that the only non-smoothcomponent of K ∗ f is given by Kδ ∗ fδ, which is supported in a ball of radius 2δ around x0.Since δ was arbitrary, the statement follows. By linearity, this strongly suggests that the localstructure of the singularities of K ∗ f can be described completely by only using knowledgeon the local structure of the singularities of f . It also suggests that the “singular part” of theoperator K should be local, with the non-local parts of K only contributing to the “regularpart”.

This discussion suggests that we certainly need the following ingredients to build anoperator K with the desired properties:• The canonical polynomial structure should be part of our regularity structure in order to

be able to describe the “regular parts”.• We should be given an “abstract integration operator” I on T which describes how the

“singular parts” ofRf transform under convolution by K.• We should restrict ourselves to models which are “compatible” with the action of I in

the sense that the behaviour of ΠxIτ should relate in a suitable way to the behaviour ofK ∗Πxτ near x.

One way to implement these ingredients is to assume first that our model space T containsabstract polynomials in the following sense.

Assumption 5.1 There exists a sector T ⊂ T isomorphic to the space of abstract polynomialsin d commuting variables. In other words, Tα 6= 0 if and only if α ∈ N, and one can find basisvectors Xk of T|k| such that every element Γ ∈ G acts on T by ΓXk = (X − h)k for someh ∈ Rd.

Furthermore, we assume that there exists an abstract integration operator I with thefollowing properties.

Assumption 5.2 There exists a linear map I : T → T such that ITα ⊂ Tα+β , such thatIT = 0, and such that, for every Γ ∈ G and τ ∈ T , one has

ΓIτ − IΓτ ∈ T . (5.1)

Finally, we want to consider models that are compatible with this structure for a givenkernel K. For this, we first make precise what we mean exactly when we said that K isapproximately homogeneous of degree β − d.

Assumption 5.3 One can write K =∑

n≥0Kn where each of the kernels Kn : Rd → R issmooth and compactly supported in a ball of radius 2−n around the origin. Furthermore, weassume that for every multiindex k, one has a constant C such that the bound

supx|DkKn(x)| ≤ C2n(d−β+|k|) , (5.2)

holds uniformly in n. Finally, we assume that∫Kn(x)P (x) dx = 0 for every polynomial P of

degree at most N , for some sufficiently large value of N .

Remark 5.4 It turns out that in order to define the operator K on Dγ , we will need K toannihilate polynomials of degree N for some N ≥ γ + β.

18 SCHAUDER ESTIMATES AND ADMISSIBLE MODELS

Remark 5.5 The last assumption may appear to be extremely stringent at first sight. Inpractice, this turns out not to be a problem at all. Say for example that we want to define anoperator that represents convolution with G, the Green’s function of the Laplacian. Then, Gcan be decomposed into a sum of terms satisfying the bound (5.2) with β = 2, but it does ofcourse not annihilate generic polynomials and it is not supported in the ball of radius 1.

However, for any fixed value ofN > 0, it is straightforward to decompose G as G = K+R,where the kernel K is compactly supported and satisfies all of the properties mentioned above,and the kernel R is smooth. Lifting the convolution with R to an operator from Dγ → Dγ+β

(actually to Dγ for any γ > 0) is straightforward, so that we have reduced our problem to thatof constructing an operator describing the convolution by K.

Given such a kernel K, we can now make precise what we meant earlier when we said thatthe models under consideration should be compatible with the kernel K.

Definition 5.6 Given a kernel K as in Assumption 5.3 and a regularity structure T satisfyingAssumptions 5.1 and 5.2, we say that a model (Π,Γ) is admissible if the identities

(ΠxXk)(y) = (y − x)k , ΠxIτ = K ∗Πxτ −ΠxJ (x)τ , (5.3)

holds for every τ ∈ T with |τ | ≤ N . Here, J (x) : T → T is the linear map given onhomogeneous elements by

J (x)τ =∑

|k|<|τ |+β

Xk

k!

∫D(k)K(x− y) (Πxτ)(dy) . (5.4)

Remark 5.7 Note first that if τ ∈ T , then the definition given above is coherent as long as|τ | < N . Indeed, since Iτ = 0, one necessarily has ΠxIτ = 0. On the other hand, theproperties of K ensure that in this case one also has K ∗Πxτ = 0, as well as J (x)τ = 0.

Remark 5.8 While K ∗ ξ is well-defined for any distribution ξ, it is not so clear a prioriwhether the operator J (x) given in (5.4) is also well-defined. It turns out that the axioms of amodel do ensure that this is the case. The correct way of interpreting (5.4) is by

J (x)τ =∑

|k|<|τ |+β

∑n≥0

Xk

k!(Πxτ)(D(k)Kn(x− ·)) .

Note now that the scaling properties of the Kn ensure that 2(β−|k|)nD(k)Kn(x − ·) is a testfunction that is localised around x at scale 2−n. As a consequence, one has

|(Πxτ)(D(k)Kn(x− ·))| . 2(|k|−β−|τ |)n ,

so that this expression is indeed summable as long as |k| < |τ |+ β.

Remark 5.9 As a matter of fact, it turns out that the above definition of an admissible modeldovetails very nicely with our axioms defining a general model. Indeed, starting from anyregularity structure T , any model (Π,Γ) for T , and a kernel K satisfying Assumption 5.3,it is usually possible to build a larger regularity structure T containing T (in the “obvious”sense that T ⊂ T and the action of G on T is compatible with that of G) and endowed with

SCHAUDER ESTIMATES AND ADMISSIBLE MODELS 19

an abstract integration map I, as well as an admissible model (Π, Γ) on T which reduces to(Π,Γ) when restricted to T . See [Hai14] for more details.

The only exception to this rule arises when the original structure T contains some homoge-neous element τ which does not represent a polynomial and which is such that |τ |+ β ∈ N.Since the bounds appearing both in the definition of a model and in Assumption 5.3 areonly upper bounds, it is in practice easy to exclude such a situation by slightly tweaking thedefinition of either the exponent β or of the original regularity structure T .

With all of these definitions in place, we can finally build the operator K : Dγ → Dγ+β

announced at the beginning of this section. Recalling the definition of J from (5.4), we set

(Kf)(x) = If (x) + J (x)f (x) + (N f)(x) , (5.5)

where the operator N is given by

(N f)(x) =∑|k|<γ+β

Xk

k!

∫D(k)K(x− y) (Rf −Πxf (x))(dy) . (5.6)

Note first that thanks to the reconstruction theorem, it is possible to verify that the righthand side of (5.6) does indeed make sense for every f ∈ Dγ in virtually the same way as inRemark 5.8. One has:

Theorem 5.10 LetK be a kernel satisfying Assumption 5.3, let T = (A, T,G) be a regularitystructure satisfying Assumptions 5.1 and 5.2, and let (Π,Γ) be an admissible model for T .Then, for every f ∈ Dγ with γ ∈ (0, N − β) and γ + β 6∈ N, the function Kf defined in (5.5)belongs to Dγ+β and satisfiesRKf = K ∗ Rf .

Proof. The complete proof of this result can be found in [Hai14] and will not be given here.Let us simply show that one has indeedRKf = K ∗Rf in the particular case when our modelconsists of continuous functions so that Remark 2.12 applies. In this case, one has

(RKf)(x) = (Πx(If (x) + J (x)f (x)))(x) + (Πx(N f)(x))(x) .

As a consequence of (5.3), the first term appearing in the right hand side of this expression isgiven by

(Πx(If (x) + J (x)f (x)))(x) = (K ∗Πxf (x))(x) .

On the other hand, the only term contributing to the second term is the one with k = 0 (whichis always present since γ > 0 by assumption) which then yields

(Πx(N f)(x))(x) =

∫K(x− y) (Rf −Πxf (x))(dy) .

Adding both of these terms, we see that the expression (K ∗Πxf (x))(x) cancels, leaving uswith the desired result.

20 APPLICATION OF THE THEORY TO SEMILINEAR SPDES

6 Application of the theory to semilinear SPDEs

Let us now briefly explain how this theory can be used to make sense of solutions to verysingular semilinear stochastic PDEs. We will keep the discussion in this section at a veryinformal level without attempting to make mathematically precise statements. The interestedreader may find more details in [Hai14].

For definiteness, we focus on the case of the dynamical Φ43 model, which is formally given

by∂tΦ = ∆Φ− Φ3 + ξ , (6.1)

where ξ denotes space-time white noise and the spatial variable takes values in the 3-dimensio-nal torus. The problem with such an equation is that even the solution to the linear part of theequation, namely

∂tΨ = ∆Ψ + ξ ,

only admits solutions in some spaces of Schwartz distributions. As a matter of fact, one hasΨ(t, ·) ∈ C−α if and only if α > 1/2. As a consequence, it turns out that the only way ofgiving meaning to (6.1) is to “renormalise” the equation by adding an “infinite” linear term“∞Φ” which counteracts the strong dissipativity of the term −Φ3. To be slightly more precise,one can prove a statement of the following kind:

Theorem 6.1 Consider the sequence of equations

∂tΦε = ∆Φε + CεΦε − Φ3ε + ξε , (6.2)

where ξε = δε ∗ξ with δε(t, x) = ε−5%(ε−2t, ε−1x), for some smooth and compactly supportedfunction %, and ξ denotes space-time white noise. Then, there exists a choice of constants Cεsuch that the sequence Φε converges in probability to a limiting (distributional) process Φ.Furthermore, the limiting process Φ does not depend on the choice of mollifier %.

Remark 6.2 It turns out that in order to obtain a limit that is independent of the choice ofmollifier %, one should take Cε of the form

Cε =c1

ε+ c log ε+ c3 ,

where c is universal, but c1 and c3 depend on the choice of %.

Remark 6.3 The limiting solution Φ is only local in time, so that the precise statement has tobe slightly tweaked to allow for finite-time blow-ups. Regarding the initial condition, one cantake Φ0 ∈ C−β for any β < 2/3. This is expected to be optimal, even for the deterministicequation.

The aim of this section is to sketch how the theory of regularity structures can be used toobtain this kind of convergence results. First of all, we note that while our solution Φ willbe a space-time distribution (or rather an element of Dγ for some regularity structure with amodel over R4), the “time” direction has a different scaling behaviour from the three “space”directions. As a consequence, it turns out to be effective to slightly change our definition of“localised test functions” by setting

ϕλ(s,x)(t, y) = λ−5ϕ(λ−2(t− s), λ−1(y − x)) .

APPLICATION OF THE THEORY TO SEMILINEAR SPDES 21

Accordingly, the “effective dimension” of our space-time is actually 5, rather than 4. Thetheory presented above extends mutatis mutandis to this setting. (Note in particular that whenconsidering the degree of a regular monomial, powers of the time variable should now becounted double.) Note also that with this way of measuring regularity, space-time white noisebelongs to C−α for every α > 5

2 . This is because of the bound

(E〈ξ, ϕλx〉2)1/2 = ‖ϕλx‖L2 ≈ λ−52 ,

combined with an argument similar to the proof of Kolmogorov’s continuity lemma.

6.1 Construction of the associated regularity structureOur first step is to build a regularity structure that is sufficiently large to allow to reformulate(6.1) as a fixed point in Dγ for some γ > 0. Denoting by G the heat kernel (i.e. the Green’sfunction of the operator ∂t −∆), we can write the solution to (6.1) with initial condition Φ0 as

Φ = G ∗ (ξ − Φ3) + GΦ0 ,

where ∗ denotes space-time convolution and where we denote by GΦ0 the harmonic extensionof Φ0. In order to have a chance of fitting this into the framework described above, we firstdecompose the heat kernel G as

G = K + K ,

where the kernel K satisfies all of the assumptions of Section 5 (with β = 2) and the remainderK is smooth. If we consider any regularity structure containing the usual Taylor polynomialsand equipped with an admissible model, is straightforward to associate to K an operatorK : Dγ → D∞ via

(Kf)(z) =∑k

Xk

k!(D(k)K ∗ Rf)(z) ,

where z denotes a space-time point and k runs over all possible 4-dimensional multiindices.Similarly, the harmonic extension of Φ0 can be lifted to an element in D∞ which we denoteagain by GΦ0 by considering its Taylor expansion around every space-time point. At this stage,we note that we actually cheated a little: while GΦ0 is smooth in {(t, x) : t > 0, x ∈ T3}and vanishes when t < 0, it is of course singular on the time-0 hyperplane {(0, x) : x ∈ T3}.This problem can be cured by introducing weighted versions of the spaces Dγ allowing forsingularities on a given hyperplane. A precise definition of these spaces and their behaviourunder multiplication and the action of the integral operator K can be found in [Hai14]. For thepurpose of the informal discussion given here, we will simply ignore this problem.

This suggests that the “abstract” formulation of (6.1) should be given by

Φ = K(Ξ− Φ3) + K(Ξ− Φ3) + GΦ0 . (6.3)

In view of (5.5), this equation is of the type

Φ = I(Ξ− Φ3) + (. . .) , (6.4)

where the terms (. . .) consist of functions that take values in the subspace T of T spannedby regular Taylor polynomials. In order to build a regularity structure in which (6.4) can beformulated, it is natural to start with the structure given by abstract polynomials (again with

22 APPLICATION OF THE THEORY TO SEMILINEAR SPDES

the parabolic scaling which causes the abstract “time” variable to have homogeneity 2 ratherthan 1), and to add a symbol Ξ to it which we postulate to have homogeneity −5

2

−, where wedenote by α− an exponent strictly smaller than, but arbitrarily close to, the value α.

We then simply add to T all of the formal expressions that an application of the right handside of (6.4) can generate for the description of Φ, Φ2, and Φ3. The homogeneity of a givenexpression is completely determined by the rules |Iτ | = |τ |+ 2 and |τ τ | = |τ |+ |τ |. Moreprecisely, we consider a collection U of formal expressions which is the smallest collectioncontaining 1, X , and I(Ξ), and such that

τ1, τ2, τ3 ∈ U ⇒ I(τ1τ2τ3) ∈ U ,

where it is understood that I(Xk) = 0 for every multiindex k. We then set

W = {Ξ} ∪ {τ1τ2τ3 : τi ∈ U} ,

and we define our space T as the set of all linear combinations of elements in W . (Notethat since 1 ∈ U , one does in particular have U ⊂ W .) Naturally, Tα consists of thoselinear combinations that only involve elements inW that are of homogeneity α. It is not toodifficult to convince oneself that, for every α ∈ R,W contains only finitely many elements ofhomogeneity less than α, so that each Tα is finite-dimensional.

In order to simplify expressions later, we will use the following shorthand graphical notationfor elements ofW . For Ξ, we simply draw a dot. The integration map is then represented by adownfacing line and the multiplication of symbols is obtained by joining them at the root. Forexample, we have

I(Ξ) = , I(Ξ)3 = , I(Ξ)I(I(Ξ)3) = .

Symbols containing factors of X have no particular graphical representation, so we will forexample write XiI(Ξ)2 = Xi . With this notation, the space T is given by

T = 〈Ξ, , , , , , , Xi , 1, , , . . .〉 ,

where we ordered symbols in increasing order of homogeneity and used 〈·〉 to denote the linearspan. Given any sufficiently regular function ξ (say a continuous space-time function), there isthen a canonical way of lifting ξ to a model ιξ = (Π,Γ) for T by setting

(ΠxΞ)(y) = ξ(y) , (ΠxXk)(y) = (y − x)k ,

and then recursively by(Πxτ τ)(y) = (Πxτ)(y) · (Πxτ)(y) , (6.5)

as well as (5.3). (Note that here we used x and y as notations for generic space-time points inorder not to overload the notations.)

It turns out furthermore that there is a canonical way of building a structure group Gfor T and to also lift ξ to a family of operators Γxy, in such a way that all of the algebraicand analytic properties of an admissible model are satisfied. With such a model ιξ at hand,it follows from (6.5) and the admissibility of ιξ that the associated reconstruction operatorsatisfies the properties

RKf = K ∗ Rf , R(fg) = Rf · Rg ,

RENORMALISATION OF THE DYNAMICAL Φ43 MODEL 23

as long as all the functions to which R is applied belong to Dγ for some γ > 0. As aconsequence, applying the reconstruction operatorR to both sides of (6.3), we see that if Φsolves (6.3) then, provided that the model (Π,Γ) = ιξ was built as above starting from anycontinuous realisation ξ of the driving noise,RΦ solves the equation (6.1).

At this stage, the situation is as follows. For any continuous realisation ξ of the drivingnoise, we have factored the solution map (Φ0, ξ)→ Φ associated to (6.1) into maps

(Φ0, ξ)→ (Φ0, ιξ)→ Φ→ RΦ ,

where the middle arrow corresponds to the solution to (6.3) in some weighted Dγ-space. Theadvantage of such a factorisation is that the last two arrows yield continuous maps, even intopologies sufficiently weak to be able to describe driving noise having the lack of regularityof space-time white noise. The only arrow that isn’t continuous in such a weak topology is thefirst one. At this stage, it should be believable that a similar construction can be performed fora very large class of semilinear stochastic PDEs. In particular, the KPZ equation can also beanalysed in this framework.

Given this construction, one is lead naturally to the following question: given a sequenceξε of “natural” regularisations of space-time white noise, for example as in (6.2), do the liftsιξε converge in probably in a suitable space of admissible models? Unfortunately, unlike in thecase of the theory of rough paths where this is very often the case, the answer to this questionin the context of SPDEs is often an emphatic no. Indeed, if it were the case for the dynamicalΦ4

3 model, then one could have chosen the constant Cε to be independent of ε in (6.2), whichis certainly not the case.

7 Renormalisation of the dynamical Φ43 model

One way of circumventing the fact that ιξε does not converge to a limiting model as ε→ 0 isto consider instead a sequence of renormalised models. The main idea is to exploit the factthat our abstract definitions of a model do not impose the identity (6.5), even in situationswhere ξ itself happens to be a continuous function. One question that then imposes itselfis: what are the natural ways of “deforming” the usual product which still lead to lifts to anadmissible model? It turns out that the regularity structure whose construction was sketchedabove comes equipped with a natural finite-dimensional group of continuous transformationsR on its space of admissible models (henceforth called the “renormalisation group”), whichessentially amounts to the space of all natural deformations of the product. It then turns out thateven though ιξε does not converge, it is possible to find a sequence Mε of elements in R suchthat the sequence Mειξε converges to a limiting model (Π, Γ). Unfortunately, the elementsMε no not preserve the image of ι in the space of admissible models. As a consequence,when solving the fixed point map (6.3) with respect to the model Mειξε and inserting thesolution into the reconstruction operator, it is not clear a priori that the resultong function (ordistribution) can again be interpreted as the solution to some modified PDE. It turns out thatin our case, at least for a certain two-parameter subgroup of R, this is again the case and themodified equation is precisely given by (6.2), where Cε is some linear combination of the twoconstants appearing in the description of Mε.

There are now three questions that remain to be answered:

1. How does one construct the renormalisation group R?

24 RENORMALISATION OF THE DYNAMICAL Φ43 MODEL

2. How does one derive the new equation obtained when renormalising a model?

3. What is the right choice of Mε ensuring that the renormalised models converge?

7.1 The renormalisation groupIn order to construct R, it is essential to first have some additional knowledge of the structuregroup G for the type of regularity structures considered above. Recall that the purpose of thegroup G is to provide a class of linear maps Γ: T → T arising as possible candidates for theaction of “reexpanding” a “Taylor series” around a different point. In our case, in view of(5.3), the coefficients of these reexpansions will naturally be some polynomials in x and in theexpressions appearing in (5.4). This suggests that we should define a space T+ whose basisvectors consist of formal expressions of the type

XkN∏i=1

Jìτi , (7.1)

where N is an arbitrary but finite number, the τi are basis elements of T , and the ì ared-dimensional multiindices satisfying |ì| < |τi| + 2. (The last bound is a reflection of therestriction of the summands in (5.4) with β = 2.) The space T+ also has a natural gradedstructure T+ =

⊕T+α by setting

|J`τ | = |τ |+ 2− |`| , |Xk| = |k| ,

and by postulating that the degree of a product is the sum of the degrees. Unlike in the caseof T however, elements of T+ all have strictly positive homogeneity, except for the emptyproduct 1 which we postulate to have degree 0.

To any given admissible model (Π,Γ), it is then natural to associate linear maps fx : T+ →R by fx(Xk) = xk, fx(σσ) = fx(σ)fx(σ), and

fx(Jìτi) =

∫D(ì)K(x− y) (Πxτi)(dy) . (7.2)

It then turns out that it is possible to build a linear map ∆: T → T ⊗T+ such that if we defineFx : T → T by

Fxτ = (I ⊗ fx)∆τ , (7.3)

where I denotes the identity operator on T , then these maps are invertible and ΠxF−1x is

independent of x. Furthermore, there exists a map ∆+ : T+ → T+ ⊗ T+ such that

(∆⊗ I)∆ = (I ⊗∆+)∆ , ∆+(σσ) = ∆+σ ·∆+σ . (7.4)

With this map at hand, we can define a product ◦ on the space of linear functionals f : T+ → Rby

(f ◦ g)(σ) = (f ⊗ g)∆+σ .

If we furthermore denote by Γf the operator T associated to any such linear functional as in(7.3), the first identity of (7.4) yields the identity ΓfΓg = Γf◦g. The second identity of (7.4)furthermore ensures that if f and g are both multiplicative in the sense that f (σσ) = f (σ)f (σ),then f ◦ g is again multiplicative. It also turns out that every multiplicative linear functional fadmits a unique inverse f−1 such that f−1 ◦ f = f ◦ f−1 = e, where e : T+ → R maps every


basis vector of the form (7.1) to zero, except for e(1) = 1. The element e is neutral in the sensethat Γe is the identity operator.

It is now natural to define the structure group G associated to T as the set of all multiplica-tive linear functionals on T+, acting on T via (7.3). Furthermore, for any admissible model,one has the identity

Γxy = F−1x Fy = Γγxy , γxy = f−1

x ◦ fy .

How does all this help with the identification of a natural class of deformations for theusual product? First, it turns out that for every continuous function ξ, if we denote again by(Π,Γ) the model ιξ, then the linear map Π : T → C given by

Π = ΠyF−1y ,

which is independent of the choice of y by the above discussion, is given by

(ΠΞ)(x) = ξ(x) , (ΠXk)(x) = xk , (7.5)

and then recursively by

Πτ τ = Πτ ·Πτ , ΠIτ = K ∗Πτ .

Note that this is very similar to the definition of ιξ, with the notable exception that (5.3) isreplaced by the more “natural” identity ΠIτ = K ∗Πτ . It turns out that the knowledge ofΠ and the knowledge of (Π,Γ) are equivalent since one has Πx = ΠFx and the map Fx canbe recovered from Πx by (7.2). (This argument appears circular but it is possible to put asuitable recursive structure on T and T+ ensuring that this actually works.) Furthermore, thetranslation (Π,Γ)↔ Π actually works for any admissible model and does not at all rely onthe fact that it was built by lifting a continuous function. However, in the general case, thefirst identity in (7.5) does not of course not make any sense anymore and might fail even if thecoordinates of Π consist of continuous functions.

At this stage we note that if ξ happens to be a stationary stochastic process and Π is builtfrom ξ by following the above procedure, then Πτ is a stationary stochastic process for everyτ ∈ T . In order to define R, it is natural to consider only transformations of the space ofadmissible models that preserve this property. Since we are not in general allowed to multiplycomponents of Π, the only remaining operation is to form linear combinations. It is thereforenatural to describe elements of R by linear maps M : T → T and to postulate their action onadmissible models by Π 7→ ΠM with

ΠMτ = ΠMτ .

It is not clear a priori whether given such a map M and an admissible model (Π,Γ) there isa coherent way of building a new model (ΠM ,ΓM ) such that ΠM is the map associated to(ΠM ,ΓM ) as above. It turns out that one has the following statement:

Proposition 7.1 In the above context, for every linear map M : T → T commuting with Iand multiplication byXk, there exist unique linear maps ∆M : T → T ⊗T+ and ∆M : T+ →T+ ⊗ T+ such that if we set

ΠMx τ = (Πx ⊗ fx)∆Mτ , γMxy (σ) = (γxy ⊗ fx)∆Mσ ,

then ΠMx satisfies again (5.3) and the identity ΠM

x ΓMxy = ΠMy .


At this stage it may look like any linear map M : T → T commuting with I and multipli-cation by Xk yields a transformation on the space of admissible models by Proposition 7.1.This however is not true since we have completely disregarded the analytical bounds that everymodel has to satisfy. It is clear from Definition 2.5 that these are satisfied if and only if ΠM

x τis a linear combination of the Πxτj with |τj | ≥ |τ |. This suggests the following definition.

Definition 7.2 The renormalisation group R consists of the set of linear maps M : T → Tcommuting with I and with multiplication by Xk, such that for τ ∈ Tα and σ ∈ T+

α , one has

∆Mτ − τ ⊗ 1 ∈⊕β>α

Tα ⊗ T+ , ∆Mσ − σ ⊗ 1 ∈⊕β>α

T+α ⊗ T+ .

Its action on the space of admissible models is given by Proposition 7.1.

7.2 The renormalised equations

In the case of the dynamical Φ4 model considered in this article, it turns out that we only needa two-parameter subgroup of R to renormalise the equations. More precisely, we considerelements M ∈ R of the form M = exp(−C1L1 − C2L2), where the two generators L1 andL2 are determined by the substitution rules

L1 : 7→ 1 , L2 : 7→ 1 .

This should be understood in the sense that if τ is an arbitrary formal expression, then L1τis the sum of all formal expressions obtained from τ by performing a substitution of the type7→ 1, and similarly for L2. For example, one has

L1 = 3 , L1 = , L2 = 3 .

One then has the following result:

Proposition 7.3 The linear maps M of the type just described belong to R. Furthermore, if(Π,Γ) is an admissible model such that Πxτ is a continuous function for every τ ∈ T , thenone has the identity

(ΠMx τ)(x) = (ΠxMτ)(x) . (7.6)

Remark 7.4 Note that it it is the same value x that appears twice on each side of (7.6). It is infact not the case that one has ΠM

x τ = ΠxMτ ! However, the identity (7.6) is all we need toderive the renormalised equations.

It is now rather straightforward to show the following:

Proposition 7.5 Let M = exp(−C1L1 − C2L2) as above and let (ΠM ,ΓM ) = Mιξ forsome smooth function ξ. Let furthermore Φ be the solution to (6.3) with respect to the model(ΠM ,ΓM ). Then, the function u(t, x) = (RMΦ)(t, x) solves the equation

∂tu = ∆u− u3 + (3C1 − 9C2)u+ ξ .


Proof. By Theorem 4.3, it turns out that (6.3) can be solved in Dγ as soon as γ is a littlebit greater than 1. Therefore, we only need to keep track of its solution Φ up to terms ofhomogeneity 1. By repeatedly applying the identity (6.4), we see that the solution Φ isnecessarily of the form

Φ = + ϕ 1− − 3ϕ + 〈∇ϕ,X〉 , (7.7)

for some real-valued function ϕ and some R3-valued function∇ϕ. (Note that∇ϕ is treated asan independent function here, we certainly do not suggest that the function ϕ is differentiable!Our notation is only by analogy with the classical Taylor expansion...) Similarly, the right handside of the equation is given up to order 0 by

Ξ− Φ3 = Ξ− − 3ϕ + 3 − 3ϕ2 + 6ϕ + 9ϕ − 3〈∇ϕ, X〉 − ϕ3 1 . (7.8)

Combining this with the definition of M , it is straightforward to see that, modulo terms ofstrictly positive homogeneity, one has

M (Ξ− Φ3) = Ξ− (MΦ)3 + 3C1 + 3C1ϕ1− 9C2 − 9C2ϕ1= Ξ− (MΦ)3 + (3C1 − 9C2)MΦ .

Combining this with (7.6), the claim now follows at once.

7.3 Convergence of the renormalised modelsIt remains to argue why one expects to be able to find constants Cε1 and Cε2 such that thesequence of renormalised models M ειξε converges to a limiting model. Instead of consideringthe actual sequence of models, we only consider the sequence of stationary processes Π

ετ :=

ΠεM ετ , where Πε is associated to (Πε,Γε) = ιξε as before. Since there are general argumentsavailable to deal with all the expressions τ of positive homogeneity, we restrict ourselves tothose of negative homogeneity which, leaving out Ξ which is easy to treat, are given by

, , , , , , Xi .

For this section, some elementary notions from the theory of Wiener chaos expansions arerequired, but we will try to hide this as much as possible. At a formal level, one has the identity

Πε = K ∗ ξε = Kε ∗ ξ ,

where the kernel Kε is given by Kε = K ∗ δε. This shows that, at least formally, one has

(Πε )(z) = (K ∗ ξε)(z)2 =

∫ ∫Kε(z − z1)Kε(z − z2) ξ(z1)ξ(z2) dz1 dz2 .

Similar but more complicated expressions can be found for any formal expression τ . Thisnaturally leads to the study of random variables of the type

Ik(f ) =

∫· · ·∫f (z1, . . . , zk) ξ(z1) · · · ξ(zk) dz1 · · · dzk . (7.9)

Ideally, one would hope to have an Ito isometry of the type EIk(f )Ik(g) = 〈f sym, gsym〉,where 〈·, ·〉 denotes the L2-scalar product and f sym denotes the symmetrisation of f . This is


unfortunately not the case. Instead, one should replace the products in (7.9) by Wick products,which are formally generated by all possible contractions of the type

ξ(zi)ξ(zj) 7→ ξ(zi) � ξ(zj) + δ(zi − zj) .

If we then set

Ik(f ) =

∫· · ·∫f (z1, . . . , zk) ξ(z1) � · · · � ξ(zk) dz1 · · · dzk ,

One has indeedEIk(f )Ik(g) = 〈f sym, gsym〉 .

See [Nua95] for a more thorough description of this construction, which also goes under thename of Wiener chaos. It turns out that one has equivalence of moments in the sense that, forevery k > 0 and p > 0 there exists a constant Ck,p such that

E|Ik(f )|p ≤ Ck,p‖f sym‖p ≤ Ck,p‖f‖p ,

where the second bound comes from the fact that symmetrisation is a contraction in L2. Finally,one has EIk(f )I`(g) = 0 if k 6= `. Random variables of the form Ik(f ) for some k ≥ 0 andsome square integrable function f are said to belong to the kth homogeneous Wiener chaos.

Returning to our problem, we first argue that it should be possible to choose M in such away that Π

εconverges to a limit as ε→ 0. The above considerations suggest that one should

rewrite Πε as

(Πε )(z) = (K ∗ ξε)(z)2 =

∫ ∫Kε(z − z1)Kε(z − z2) ξ(z1) � ξ(z2) dz1 dz2 +Cε , (7.10)

where the constant Cε is given by

Cε =

∫K2ε (z1) dz1 =

∫K2ε (z − z1) dz1 .

Note now thatKε is an ε-approximation of the kernelK which has the same singular behaviouras the heat kernel. In terms of the parabolic distance, the singularity of the heat kernel scales likeK(z) ∼ |z|−3 for z → 0. (Recall that we consider the parabolic distance |(t, x)| =

√|t|+ |x|,

so that this is consistent with the fact that the heat kernel is bounded by t−3/2.) This suggeststhat one has K2

ε (z) ∼ |z|−6 for |z| � ε. Since parabolic space-time has scaling dimension 5(time counts double!), this is a non-integrable singularity. As a matter of fact, there is a wholepower of z missing to make it borderline integrable, which suggests that one has

Cε ∼1

ε.

This already shows that one should not expect Πε to converge to a limit as ε→ 0. However,it turns out that the first term in (7.10) converges to a distribution-valued stationary space-timeprocess, so that one would like to somehow get rid of this diverging constant Cε. This isexactly where the renormalisation map M (in particular the factor exp(−C1L1)) enters intoplay. Following the above definitions, we see that one has

(Πε

)(z) = (ΠεM )(z) = (Πε )(z)− C1 .


This suggests that if we make the choice C1 = Cε, then Πε

does indeed converge to anon-trivial limit as ε→ 0. This limit is a distribution given by

(Πε )(ψ) =

∫ ∫ψ(z)K(z − z1)K(z − z2) dz ξ(z1) � ξ(z2) dz1 dz2 .

Using again the scaling properties of the kernel K, it is not too difficult to show that this yieldsindeed a random variable belonging to the second homogeneous Wiener chaos for every choiceof smooth test function ψ. Once we know that Π

εconverges, it is immediate that Π

εX

converges as well, since this amounts to just multiplying a distribution by a smooth function.A similar argument to what we did for allows to take care of τ = since one then has

(Πε )(z) =

∫ ∫Kε(z − z1)Kε(z − z2)Kε(z − z3) ξ(z1) � ξ(z2) � ξ(z3) dz1 dz2 dz3

+ 3

∫ ∫Kε(z − z1)Kε(z − z2)Kε(z − z3)δ(z1 − z2) ξ(z3) dz1 dz2 dz3 .

Noting that the second term in this expression is nothing but

3Cε

∫Kε(z − z1) ξ(z1) dz1 = 3Cε(Π

ε )(z) ,

we see that in this case, provided again that C1 = Cε, Πε

is given by only the first termin the expression above, which turns out to converge to a non-degenerate limiting randomdistribution in a similar way to what happened for .

Turning to our list of terms of negative homogeneity, it remains to consider , , and .It turns out that the latter two are the more difficult ones, so we only discuss these. Let us firstargue why we expect to be able to choose the constants C1 and C2 in such a way that Π

ε

converges to a limit. In this case, the “bad” terms comes from the part of (Πε )(z) belongingto the homogeneous chaos of order 0. This is simply a constant, which turns out to be given by

Cε = 2

∫K(z)Q2

ε(z) dz , (7.11)

where the kernel Qε is given by

Qε(z) =

∫Kε(z)Kε(z − z) dz .

Since Kε is an ε-mollification of a kernel with a singularity of order −3 and the scalingdimension of the underlying space is 5, we see that Qε behaves like an ε-mollification ofa kernel with a singularity of order −3 − 3 + 5 = −1 at the origin. As a consequence,the singularity of the integrand in (7.11) is of order −5, which gives rise to a logarithmicdivergence as ε→ 0. This suggests that one should choose C2 = Cε in order to cancel out thisdiverging term and obtain a non-trivial limit for Π

εas ε→ 0. This is indeed the case.

We finally turn to the symbol . In this case, the “bad” terms appearing in the Wienerchaos decomposition of Πε are the terms in the first homogeneous Wiener chaos, which areof the form

3

∫Qε(z − z1)Kε(z1 − z2)ξ(z2) dz1 dz2 = 3

∫(Qε ∗Kε)(z − z2)ξ(z2) dz2 , (7.12)


where Qε is the kernel given by

Qε(z) = 2K(z)Q2ε(z) .

As already mentioned above, the problem here is that as ε → 0, Qε converges to a kernelQ = 2KQ2, which has a non-integrable singularity at the origin. In particular, the action ofintegrating a test function against Qε does not converge to a limiting distribution as ε→ 0.

This is akin to the problem of making sense of integration against a one-dimensionalkernel with a singularity of type 1/|x| at the origin. For the sake of the argument, let usconsider a function W : R→ R which is compactly supported and smooth everywhere exceptat the origin, where it diverges like W (x) ∼ 1/|x|. It is then natural to associate to W a“renormalised” distribution RW given by

(RW )(ϕ) =

∫W (x)(ϕ(x)− ϕ(0)) dx .

Note that RW has the property that if ϕ(0) = 0, then it simply corresponds to integrationagainst W , which is the standard way of associating a distribution to a function. In a way, theextra term can be interpreted as subtracting a Dirac distribution with an “infinite mass” locatedat the origin, thus cancelling out the divergence of the non-integrable singularity. It is alsostraightforward to verify that if Wε is a sequence of smooth approximations to W (say one hasWε(x) = W (x) for |x| > ε and Wε ∼ 1/ε otherwise), then RW ε → RW in a distributionalsense, and (using the usual correspondence between functions and distributions) one has

RW ε = W ε − Cεδ0 , Cε =

∫W ε(x) dx .

The cure to the problem we are facing for showing the convergence of Πε is virtuallyidentical. Indeed,by choosing C2 = Cε as in (7.11), the term in the first homogeneous Wienerchaos for Π

εcorresponding to (7.12) is precisely given by

3

∫Qε(z − z1)Kε(z1 − z2)ξ(z2) dz1 dz2 − 3C2

∫Kε(z − z2)ξ(z2) dz2

= 3

∫(RQε ∗Kε)(z − z2)ξ(z2) dz2 .

It turns out that the convergence of RQε to a limiting distribution RQ takes place in asufficiently strong topology to allow to conclude that Π

εdoes indeed converge to a non-

trivial limiting random distribution.It should be clear from this whole discussion that while the precise values of the constants

C1 and C2 depend on the details of the mollifier δε, the limiting (random) model (Π, Γ)obtained in this way is independent of it. Combining this with the continuity of the solution tothe fixed point map (6.3) and of the reconstruction operatorR with respect to the underlyingmodel, we see that the statement of Theorem 6.1 follows almost immediately.

References

[AR91] S. ALBEVERIO and M. ROCKNER. Stochastic differential equations in infinite dimensions:solutions via Dirichlet forms. Probab. Theory Related Fields 89, no. 3, (1991), 347–386.doi:10.1007/BF01198791.

http://dx.doi.org/10.1007/BF01198791


[BG97] L. BERTINI and G. GIACOMIN. Stochastic Burgers and KPZ equations from particle systems.Comm. Math. Phys. 183, no. 3, (1997), 571–607. doi:10.1007/s002200050044.

[BP08] J. BOURGAIN and N. PAVLOVIC. Ill-posedness of the Navier-Stokes equationsin a critical space in 3D. J. Funct. Anal. 255, no. 9, (2008), 2233–2247.doi:10.1016/j.jfa.2008.07.008.

[Dau88] I. DAUBECHIES. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl.Math. 41, no. 7, (1988), 909–996. doi:10.1002/cpa.3160410705.

[DPD03] G. DA PRATO and A. DEBUSSCHE. Strong solutions to the stochastic quantization equations.Ann. Probab. 31, no. 4, (2003), 1900–1916. doi:10.1214/aop/1068646370.

[GLP99] G. GIACOMIN, J. L. LEBOWITZ, and E. PRESUTTI. Deterministic and stochastic hydro-dynamic equations arising from simple microscopic model systems. In Stochastic partialdifferential equations: six perspectives, vol. 64 of Math. Surveys Monogr., 107–152. Amer.Math. Soc., Providence, RI, 1999.

[Gub04] M. GUBINELLI. Controlling rough paths. J. Funct. Anal. 216, no. 1, (2004), 86–140.doi:10.1016/j.jfa.2004.01.002.

[Gub10] M. GUBINELLI. Ramification of rough paths. J. Differential Equations 248, no. 4, (2010),693–721. doi:10.1016/j.jde.2009.11.015.

[Hai11] M. HAIRER. Rough stochastic PDEs. Comm. Pure Appl. Math. 64, no. 11, (2011), 1547–1585. doi:10.1002/cpa.20383.

[Hai13] M. HAIRER. Solving the KPZ equation. Ann. of Math. (2) 178, no. 2, (2013), 559–664.doi:10.4007/annals.2013.178.2.4.

[Hai14] M. HAIRER. A theory of regularity structures. Invent. Math. (2014).doi:10.1007/s00222-014-0505-4.

[JLM85] G. JONA-LASINIO and P. K. MITTER. On the stochastic quantization of field theory. Comm.Math. Phys. 101, no. 3, (1985), 409–436.

[KPZ86] M. KARDAR, G. PARISI, and Y.-C. ZHANG. Dynamic scaling of growing interfaces. Phys.Rev. Lett. 56, no. 9, (1986), 889–892.

[LCL07] T. J. LYONS, M. CARUANA, and T. LEVY. Differential equations driven by rough paths,vol. 1908 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Lectures from the34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With anintroduction concerning the Summer School by Jean Picard.

[Lyo98] T. J. LYONS. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14,no. 2, (1998), 215–310.

[Mey92] Y. MEYER. Wavelets and operators, vol. 37 of Cambridge Studies in Advanced Mathematics.Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original byD. H. Salinger.

[Nua95] D. NUALART. The Malliavin calculus and related topics. Probability and its Applications(New York). Springer-Verlag, New York, 1995.

[PW81] G. PARISI and Y. S. WU. Perturbation theory without gauge fixing. Sci. Sinica 24, no. 4,(1981), 483–496.

[Sim97] L. SIMON. Schauder estimates by scaling. Calc. Var. Partial Differential Equations 5, no. 5,(1997), 391–407. doi:10.1007/s005260050072.

[You36] L. C. YOUNG. An inequality of the Holder type, connected with Stieltjes integration. ActaMath. 67, no. 1, (1936), 251–282.

http://dx.doi.org/10.1007/s002200050044

http://dx.doi.org/10.1016/j.jfa.2008.07.008

http://dx.doi.org/10.1002/cpa.3160410705

http://dx.doi.org/10.1214/aop/1068646370

http://dx.doi.org/10.1016/j.jfa.2004.01.002

http://dx.doi.org/10.1016/j.jde.2009.11.015

http://dx.doi.org/10.1002/cpa.20383

http://dx.doi.org/10.4007/annals.2013.178.2.4

http://dx.doi.org/10.1007/s00222-014-0505-4

http://dx.doi.org/10.1007/s005260050072

Date post:	25-Mar-2018
Category:	Documents
Upload:	doannga
View:	218 times
Download:	1 times

Introduction to Regularity · PDF fileINTRODUCTION 3 From a “philosophical”...

Documents